3.1 Hydraulic circuit analysis [hydraulic circuits top]

Recall from Chapter 2 that the solution for steady flow of a Newtonian fluid through a channel with uniform circular cross-section aligned in the z-direction and defined by a radius R is given by

From this velocity distribution, we also determined that the volumetricflowrate Q is given by:

Strictly speaking, this result applies only for a channel that is infinitely long. Let us, though, neglect entrance effects and simply use the infinite-tube result as an approximation of the flow through a finite tube. This approximation is good if Re ≪ 1, in which case we can assume that  is uniform along the length of the tube, andthus we can write = Δp∕L, where Δp is the difference between the pressure at the exit and that at the inlet, and L is the length of the tube. With this approximation, we have

(3.3)

which can be rewritten as

(3.4)

If we now definethe hydraulic resistance as

(3.5)

we can write Equation 3.2 as

This relation is the Hagen-Poiseuille law, withthe hydraulic resistance in Equation 3.5 defined for a channel of circular cross-section and radius R. This relation is approximately correct for long channels (Re ≪ 1), for which this relation’s errors (which are primarily at the entrance to the tube) can be neglected, and furthermore requires that the flowrate be laminar (Re < 2300). Equation 3.6 is analogous to Ohm’s law for electrical circuits (I = ΔV∕R), which is presented in Chapter 5, and we adopt notation and symbolism similar to that for electrical circuits (Figure 3.1) when denoting a channel in a channel network. In fact, all of the relations for hydraulic circuits are directly analogous to electrical circuit relations, and those familiar with electrical circuits typically find these relations intuitive.

Now consider several tubes that intersect at a rigid node or junction. Conservation of massrequires that the net flow into or out of the junction must be zero:

where positive Q denotes flowrate into the node.

The Hagen-Poiseuille law and conservation of mass together form the basis for steady-state hydraulic circuit analysis. Because both of these relations are linear relations, the pressures and flowrates in a fluid network can be written symbolically (see Figure 3.2) and determined using algebraic equations. In the sections below, we use this approach to describe flow through channels of a variety of cross-sections, and furthermore account for channels that are not rigid.

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